Stability for semidiscrete Galerkin approximation of neutral delay equations

Author(s):  
R.H. Fabiano ◽  
J. Turi
Author(s):  
Zaid Ahsan ◽  
Thomas K. Uchida ◽  
Akash Subudhi ◽  
C. P. Vyasarayani

Falling is the leading cause of both fatal and nonfatal injury in the elderly, often requiring expensive hospitalization and rehabilitation. We study the stability of human balance during stance using inverted single- and double-pendulum models, accounting for physiological reflex delays in the controller. The governing second-order neutral delay differential equation (NDDE) is transformed into an equivalent partial differential equation (PDE) constrained by a boundary condition and then into a system of ordinary differential equations (ODEs) using the Galerkin method. The stability of the ODE system approximates that of the original NDDE system; convergence is achieved by increasing the number of terms used in the Galerkin approximation. We validate our formulation by deriving analytical expressions for the stability margins of the double-pendulum human stance model. Numerical examples demonstrate that proportional–derivative–acceleration (PDA) feedback generally, but not always, results in larger stability margins than proportional–derivative (PD) feedback in the presence of reflex delays.


2006 ◽  
Vol 2006 ◽  
pp. 1-9 ◽  
Author(s):  
Xiao-Bao Shu ◽  
Yuan-Tong Xu

By means of a variational structure andZ2-group index theory, we obtain multiple periodic solutions to a class of second-order nonlinear neutral delay equations of the formx″(t−τ)+λ(t)f(t,x(t),x(t−τ),x(t−2τ))=x(t),λ(t)>0,τ>0.


2008 ◽  
Vol 2008 ◽  
pp. 1-13 ◽  
Author(s):  
M. Ayachi ◽  
J. Blot

We provide new variational settings to study the a.p. (almost periodic) solutions of a class of nonlinear neutral delay equations. We extend Shu and Xu (2006) variational setting for periodic solutions of nonlinear neutral delay equation to the almost periodic settings. We obtain results on the structure of the set of the a.p. solutions, results of existence of a.p. solutions, results of existence of a.p. solutions, and also a density result for the forced equations.


2014 ◽  
Vol 69 (4) ◽  
pp. 1027-1056 ◽  
Author(s):  
M. V. Barbarossa ◽  
K. P. Hadeler ◽  
C. Kuttler

1997 ◽  
Vol 137 (2) ◽  
pp. 320-339 ◽  
Author(s):  
A.A Al-Nayef ◽  
P.E Kloeden ◽  
A.V Pokrovskii

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